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We study the convergence rate of solutions to the initial-boundary value problemfor scalar viscous conservation laws on the half line. Especially, we deal with the case where theRiemann problem for the corresponding hyperbolic equation admits transonic rarefaction waves. Inthis case, it is known that the solution tends toward a linear superposition of the stationary solutionand the rarefaction wave. We show that the convergence rate is (1 + t).12 (1. 1p ) log2(2 + t) in Lpnorm (1  p < 1) and (1 + t).12+. in L1 norm if the initial perturbation from the correspondingsuperposition is located in H1 \ L1. The proof is given by a combination of the weighted Lp energymethod and the L1 estimate.

Convergence Rate to the Nonlinear Waves for Viscous Conservation Laws on the Half Line